# Solution of a tridiagonal system of linear equations

### Description

Solves the linear system of equations

*Ax=B*

where A has to be square and *tridiagonal*, i.e with nonzero elements
only on, one band above, and one band below the diagonal.

### Usage

1 | ```
Solve.tridiag ( diam1, dia, diap1, B=rep(0,times=length(dia)))
``` |

### Arguments

`diam1 ` |
a vector with (nonzero) elements below the diagonal. |

`dia ` |
a vector with (nonzero) elements on the diagonal. |

`diap1 ` |
a vector with (nonzero) elements above the diagonal. |

`B ` |
Right-hand side of the equations, a vector with length = number of rows of A, or a matrix with number of rows = number of rows of A. |

### Details

If the length of the vector `dia`

is equal to N, then the lengths of
`diam1`

and `diap1`

should be equal to N-1

### Value

matrix with the solution, `X`

, of the tridiagonal system of equations Ax=B.
The number of columns of this matrix equals the number of columns of B.

### Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

### See Also

`Solve.banded`

, the function to solve a banded system of
linear equations.

`Solve.block`

, the function to solve a block diagonal system of
linear equations.

`Solve`

the generalised inverse solution,

`solve`

the R default

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
# create tridagonal system: bands on diagonal, above and below
nn <- 20 # nr rows and columns of A
aa <- runif(nn)
bb <- runif(nn)
cc <- runif(nn)
# full matrix
A <- matrix(nrow = nn, ncol = nn, data = 0)
diag(A) <- bb
A[cbind(1:(nn-1), 2:nn)] <- cc[-nn]
A[cbind(2:nn, 1:(nn-1))] <- aa[-1]
B <- runif(nn)
# solve as full matrix
solve(A, B)
# same, now using tridiagonal algorithm
as.vector(Solve.tridiag(aa[-1], bb, cc[-nn], B))
# same, now with 3 different right hand sides
B3 <- cbind(B, B*2, B*3)
Solve.tridiag(aa[-1], bb, cc[-nn], B3)
``` |